^{1,a)}

### Abstract

We investigate the difficulties that students in calculus-based introductory physics courses have with the concepts of symmetry, electric field, and electric flux which are important for applying Gauss’s law. The determination of the electric field using Gauss’s law requires determining the symmetry of a particular charge distribution and predicting the direction of the electric field everywhere if a high symmetry exists. Effective application of Gauss’s law implicitly requires understanding the principle of superposition for electric fields. Helping students learn when Gauss’s law can be readily applied to determine the strength of the electric field, and then helping them learn to determine the appropriate shape of Gaussian surfaces if sufficient symmetry exists, can help develop their reasoning and problem-solving skills. We administered free-response and multiple-choice questions and conducted interviews with individual students using a think-aloud protocol to elucidate the difficulties that students have with the concepts of symmetry, electric field, and electric flux. We also developed a multiple-choice test that targets these conceptual issues to obtain quantitative information about their difficulties and administered it to 541 students in the introductory calculus-based physics courses and to upper-level undergraduates in an electricity and magnetism course and to graduate students enrolled in a teaching assistant seminar course. We find that undergraduate students have many common difficulties with these concepts.

We are very grateful to P. Reilly and all the faculty who reviewed the various components of the test at various stages and provided invaluable feedback. We are also very thankful to all the faculty who administered the test. We thank R. Devaty, A. Janis, R. Johnsen, and J. Levy for a critical reading of the manuscript. This work is supported in part by the National Science Foundation Award No. DUE-0442087.

I. INTRODUCTION

II. PREVIOUS INVESTIGATIONS RELATED TO ELECTRICITY AND MAGNETISM

III. METHOD

IV. BACKGROUND

V. DISCUSSION OF STUDENT DIFFICULTIES

A. The electric charge and electric flux are scalars

B. The principle of superposition

C. The electric field inside a hollow nonconducting object

D. The underlying symmetry of a charge distribution

E. Electric field and electric flux

F. Other difficulties

G. Performance of upper-level undergraduates

H. Performance of graduate students

VI. SUMMARY

### Key Topics

- Electric fields
- 92.0
- Surface charge
- 30.0
- Graduates
- 16.0
- Double layers
- 13.0
- Physics education
- 12.0

## Figures

Charge distribution for Problem 2.

Charge distribution for Problem 2.

A horizontal square sheet of charge for Problem 3.

A horizontal square sheet of charge for Problem 3.

Charge distribution for Problem 4.

Charge distribution for Problem 4.

Charge distribution for Problem 7.

Charge distribution for Problem 7.

Gaussian surfaces for Problem 9.

Gaussian surfaces for Problem 9.

Diagram for Problem 10.

Diagram for Problem 10.

Diagram for Problem 11.

Diagram for Problem 11.

Diagram for Problem 12.

Diagram for Problem 12.

Diagram for Problems 14 and 15.

Diagram for Problems 14 and 15.

Diagram for Problem 16.

Diagram for Problem 16.

Diagram for Problem 17.

Diagram for Problem 17.

Diagram for Problems 19 and 20.

Diagram for Problems 19 and 20.

Diagram for Problems 21 and 22.

Diagram for Problems 21 and 22.

Diagram for Problem 23.

Diagram for Problem 23.

Diagram for Problem 24.

Diagram for Problem 24.

Diagram for Problem 25.

Diagram for Problem 25.

## Tables

Percentage of introductory calculus-based physics students (total number of students 541) who selected choices (a)–(e) on Problems (1)–(25) on the test. The correct response for each question has been italicized. The average score was 49%.

Percentage of introductory calculus-based physics students (total number of students 541) who selected choices (a)–(e) on Problems (1)–(25) on the test. The correct response for each question has been italicized. The average score was 49%.

Concepts covered and the questions that addressed them in the multiple-choice test.

Concepts covered and the questions that addressed them in the multiple-choice test.

Percentage of students in the upper-level undergraduate E&M course (total number of students 33) who selected choices (a)–(e) on the pretest (before instruction in the upper-level course). The correct response for each question is italicized. The average score was 44%.

Percentage of students in the upper-level undergraduate E&M course (total number of students 33) who selected choices (a)–(e) on the pretest (before instruction in the upper-level course). The correct response for each question is italicized. The average score was 44%.

Percentage of students in the upper-level undergraduate E&M course (total number of students 28) who selected the choices (a)–(e) on the post-test (after instruction in the upper-level course). The correct response for each question is italicized. The average score was 49%.

Percentage of students in the upper-level undergraduate E&M course (total number of students 28) who selected the choices (a)–(e) on the post-test (after instruction in the upper-level course). The correct response for each question is italicized. The average score was 49%.

Percentage of physics graduate students enrolled in a course for teaching assistants (total number of students 33) who selected the choices (a)–(e) on Problems (1)–(25) on the test. The correct response for each question has been italicized. The average score was 75%.

Percentage of physics graduate students enrolled in a course for teaching assistants (total number of students 33) who selected the choices (a)–(e) on Problems (1)–(25) on the test. The correct response for each question has been italicized. The average score was 75%.

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